Lectures on Quantum Field Theory. Ashok Das
case, W0, indeed represents the helicity operator up to a normalization as we had noted in (4.113). It follows now from (4.115) that (the contributions from W0 and W3 cancel out)
Let us now determine the dimensionality of the massless representations algebraically. Let us recall that we are considering a massless state with momentum of the form pµ = (p, 0, 0, −p) and we would like to determine the “little” group of symmetries associated with such a vector. We recognize that in this case, the set of Lorentz transformations which would leave this four vector invariant must include rotations around the z-axis. This can be seen intuitively from the fact that the motion of the particle is along the z axis, but also algebraically by recognizing that a light-like vector of the form being considered is an eigenstate of the operator P0 − P3, namely,
Furthermore, from the Poincaré algebra in (4.38), we see that
so that rotations around the z-axis define a symmetry of the light-like vector (state) that we are considering. To determine the other symmetries of a light-like vector, let us define two new operators as
It follows now that these operators commute with P0 − P3 in the space of light-like states, namely,
and, therefore, also define symmetries of light-like states. These represent all the symmetries of the light-like vector (state). We note that the algebra of the symmetry generators takes the form
Namely, it is isomorphic to the algebra of the Euclidean group in two dimensions, E2 (which consists of translations and rotation). Thus, we say that the stability group or the “little” group of a light-like vector is E2. Clearly, M12 is the generator of rotations around the z axis or in the two dimensional plane and Π1, Π2 have the same commutation relations as those of translations in this two dimensional space. Furthermore, comparing with Wi, i = 1, 2, 3 in (4.115), we see that up to a normalization, the three independent Pauli-Lubanski operators are, in fact, the generators of symmetry of the “little” group, as we had also seen in the massive case. This may seem puzzling, but can be easily understood as follows. We note from (4.90) that in the momentum basis states (where pµ is a number), the Pauli-Lubanski operators satisfy an algebra and, therefore, can be thought of as generators of some transformations. The meaning of the transformations, then, follows from (4.86) as the transformations that leave pµ invariant. Namely, they generate transformations which will leave the momentum basis states invariant. This is, of course, what we have been investigating within the context of “little” groups.
Let us note from (4.121) that Π1 ∓ iΠ2 correspond respectively to raising and lowering operators for M12, namely,
Let us also note for completeness that the Casimir of the E2 algebra is given by
and comparing with (4.116), we see that in the space of light-like momentum states W2 ∝ Π2. Since Π1, Π2 correspond to generators of “translation”, their eigenvalues can take any value. As a result, if W2 ≠ 0 in this space, we note from (4.122) that spin can take an infinite number of values which, as we have already pointed out, does not correspond to any physical system. On the other hand, if W2 = 0 in this space of states, then it follows from (4.123) that (h corresponds to the helicity quantum number)
(Alternatively, we can say that Π1|p, h〉 = 0 = Π2|p, h〉 and this is the reason for the earlier assertion.) This corresponds to the one dimensional representation of E2 known as the “degenerate” representation. Clearly, such a state would correspond to the highest or the lowest helicity state. Furthermore, if our theory is also invariant under parity (or space reflection), the space of physical states would also include the state with the opposite helicity (recall that helicity changes sign under space reflection, see (3.148)). As a result, massless theories with nontrivial spin that are parity invariant would have two dimensional representations corresponding to the highest and the lowest helicity states, independent of the spin of the particle. On the other hand, if the theory is not parity invariant, the dimensionality of the representation will be one dimensional, as we have seen explicitly in the case of massless fermion theories describing neutrinos.
Incidentally, the fact that the massless representations have to be one dimensional, in general, can be seen in a heuristic way as follows. Let us consider spin as arising from a circular motion. Then, it is clear that since a massless particle moves at the speed of light, the only consistent circular motion a massless particle can have, is in a plane perpendicular to the direction of motion (otherwise, some component of the velocity would exceed the speed of light). In other words, in such a case, spin can only be either parallel or anti-parallel to the direction of motion leading to the one dimensional nature of the representation. However, if parity (space reflection) is a symmetry of the system, then we must have states corresponding to both the circular motions leading to the two dimensional representation.
4.4References
1.V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals of Mathematics 48, 568 (1947).
2.A. Das and S. Okubo, Lie Groups and Lie Algebras for Physicists, Hindustan Publishing, India and World Scientific Publishing, Singapore (2014).
3.E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of Mathematics 40, 149 (1939).
4.E. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York (1959).
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1See, for example, Quantum Mechanics: A Modern Introduction, A. Das and A. C. Melissinos (Gordon and Breach), page 289 or Lectures on Quantum Mechanics, A. Das (Hindustan Book Agency, New Delhi), page 182 (note there is a typo in the sign of the 23 element for L2